A note on the stratified domination number of generalized planar Petersen like graphs

Year 2020, Volume: 2 Issue: 1, - , 01.01.2020

Mehmet Şerif Aldemir

Abstract

Let G be a graph with the vertex set V (G). G is called 2-stratified if V (G) is partitioned into red and blue vertices. The stratified domination number of a graph G is the minimum number of red vertices of V (G) in a red-blue coloring of the vertices of V (G) such that every blue vertex v of V (G) lies in a vuw (blue, blue, red) path in G for a blue vertex u ∈ V (G) (u 6= v) and a red vertex w ∈ V (G). In this paper we first define the concept of generalized planar Petersen like graphs PP(n,2) for any positive odd integers and study the stratified domination number of generalized planar Petersen like graphs

Keywords

stratified domination, 2-stratified graphs

References

• [1] R. Rashidi, The Theory and Applications of Stratified Graphs (Ph.D. Dissertation, Western Michigan University, 1994).
• [2] G. Chartrand, H. Gavlas, M.A. Henning and R. Rashidi, Stratidistance instratified graphs, Math. Bohem. 122 (1997) 337-347.
• [3] G. Chartrand, L. Holley, R. Rashidi, N.A. Sherwani, Distance in stratifiedgraphs, Czech. Math. J. 125 (2000) 135-146.
• [4] G. Chartrand, T.W. Haynes, M.A. Henning, P. Zhang, Stratification anddomination in graphs, Discrete Math. 272 (2003) 171-185.
• [5] G. Chartrand, T.W. Haynes, M.A. Henning, P. Zhang, Stratified claw domination in prisms, J. Combin. Math. Combin. Comput. 33 (2000) 81-96.
• [6] R. Gera, P. Zhang, Bounds for the F-domination number of a graph, Congr.Numer. 166 (2004) 131-144.
• [7] R. Gera, P. Zhang, Realizable triples for stratified domination in graphs,Math. Bohem. 130 (2005) 185-202.
• [8] R. Gera, P. Zhang, On stratified domination in oriented graphs, Congr.Numer. 173 (2005) 175-192.
• [9] R. Gera, P. Zhang, On stratification and domination in graphs, Discuss.Math. Graph Theory 26 (2006) 249-272.
• [10] R. Gera, P. Zhang, Stratified domination in oriented graphs, J. Combin.Math. Combin. Comput. 60 (2007) 105-125.
• [11] M.A. Henning, J.E. Maritz, Stratification and domination in graphs II,Discrete Math. 286 (2004) 203-211.
• [12] M.A. Henning, J.E. Maritz, Stratification and domination in graphs withminimum degree two, Discrete Math. 301 (2005) 175-194.
• [13] M.A. Henning, J.E. Maritz, Stratification and domination in graphs withminimum degree two, Discrete Math. 301 (2005) 175-194.
• [14] M.A. Henning, J.E. Maritz, Stratification and domination in prisms, ArsCombin. 81 (2006) 343-358.
• [15] M.A. Henning, J.E. Maritz, Simultaneous stratification and domination ingraphs with minimum degree two, Quaest. Math. 29 (2006) 1-16.
• [16] T.W. Haynes, M.A. Henning, P. Zhang, A survey of stratified dominationin graphs, Discrete Math. 309 (2009) 5806-5819.
• [17] G. J. Chang, C. W. Chang, D. Kuo, S.H. Poon, Algorithmic aspect ofstratified domination in graphs, Information Processing Letters 113 (2013)861-865
• [18] C. W. Chang, D. Kuo, S.C. Liaw and J.H. Yan, F3-domination problemsin graphs, Journal of Combinatorial Optimization 28 (2014) 400-413.
• [19] J. Liu, X. Zhang, The exact domination number of generalized Petersengraphs, Comp. Appl. Math. (2014) 33:497-506.
• [20] W. S. Li, H.M. Xing, M. Y. Sohn, On the signed total domination numberof generalized Petersen graphs P(n,2), Bull. Korean Math. Soc. 50 (2013)2021-2026.
• [21] X. Fu, Y. Yang, B. Jiang, On the domination number of generalized Petersen graphs P(n,2), Discrete Mathematics 309 (2009) 2445-2451.
• [22] H. Wang, X. Xu, Y. Yang, K. Lu¨, Liar’s domination number of generalizedPetersen graphs P(n,1) and P(n,2), Util. Math. 82 (2012) 317-335.
• [23] R. Barrera, D. Ferrero, Power domination in cylinders, tori, and generalized Petersen graphs, Networks 58 (2011) 43-49.
• [24] Z. Shao, H. Jiang, P. Wu, S. Wang, J. erovnik, X. Zhang, J.B Liu, On 2rainbow domination of generalized Petersen graphs. Discrete Appl. Math.257 (2019), 370384.